Question

How many ways can you make a sandwich by choosing $4$ out of $10$ ingredients?

Answer 1

Here we must proceed by just knowing the simple fact which says that when we need to choose $n$ number of things out of $m$ things then we can select it in ${}^m{C_n}$ ways and this is only what is asked in the above problem. So we just need to know how we can calculate ${}^m{C_n}$ value.

Complete step by step solution:
Here we are given that we need to make a sandwich by choosing or selecting $4$ out of $10$ ingredients and therefore we must know that whenever we have such problem where we need to select the $n$ number of things out of $m$ things then we can select it in ${}^m{C_n}$ ways
Now we must know how to calculate its value as a number. So the formula for it is:
${}^m{C_n} = \dfrac{{m!}}{{\left( {m - n} \right)!n!}}$
Here the symbol represent the factorial and its value is calculated by multiplying the number in the preceding order till $1$
For example: If we need to calculate the value of $3!$ then it will be $\left( 3 \right)\left( 2 \right)\left( 1 \right) = 6$
So we can write its general formula as:
$n! = \left( n \right)\left( {n - 1} \right)\left( {n - 2} \right)..............\left( 3 \right)\left( 2 \right)\left( 1 \right)$
Similarly here we are given that we need to choose $4$out of $10$ ingredients. Hence we can compare it with the above general formula. We come to know that $m = 10,n = 4$ and we need to calculate the value of ${}^{10}{C_4}$
Hence we can say that:
${}^{10}{C_4} = \dfrac{{10!}}{{4!\left( {10 - 4} \right)!}} = \dfrac{{10!}}{{4!6!}}$
Now we can also write $10! = \left( {10} \right)\left( 9 \right)\left( 8 \right)\left( 7 \right)\left( {6!} \right)$ and $4! = \left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right) = 24$
Substituting these values we will get:
${}^{10}{C_4} = \dfrac{{10!}}{{4!6!}} = \dfrac{{\left( {10} \right)\left( 9 \right)\left( 8 \right)\left( 7 \right)\left( {6!} \right)}}{{\left( {24} \right)\left( {6!} \right)}} = 210$

Hence we can say that we can choose $4$out of $10$ ingredients in $210{\text{ ways}}$.

Note:
Here the student is asked only to select but if we are given anytime to select as well as arrange then we need to apply the formula of ${}^m{P_n} = \dfrac{{m!}}{{\left( {m - n} \right)!}}$ where $P$ stands for permutation.